Propagation in Hamiltonian dynamics and relative symplectic homology

TL;DR

This paper proves the existence of noncontractible periodic orbits in certain Hamiltonian systems on cotangent bundles using Floer homology and relative symplectic capacity, with applications to propagation and periodic orbit problems.

Contribution

It introduces a new approach combining Floer homology and relative symplectic capacity to establish periodic orbit existence in Hamiltonian dynamics on cotangent bundles.

Findings

Existence of noncontractible periodic orbits under large Hamiltonians

Propagation properties of sequential Hamiltonian systems demonstrated

Periodic orbits on hypersurfaces and Hamiltonian circle actions established

Abstract

The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.